Abusing mathematical notation, are these examples of abuse?

The example you've given of a function is not an abuse. $x$ is instead shorthand for $\pi_1(t)$ and $y$ is shorthand for $\pi_2(t)$ and $(x,y)$ is shorthand for $t$.

$g \in G$ is a very minor abuse, yes. "A group $G$ is a set $G$ endowed with some operations" is a slight abuse, but one which will never be misinterpreted. It is done this way to avoid the proliferation of unnecessary and confusing symbols. For the same reason, we use the symbol $+$ to refer to the three different operations of addition of integers, rationals, and reals.

Suppose that in the two situations we cited, the entire mathematical community were to agree overnight to switch notation. What would we gain, besides the smug satisfaction of pedantry? The new notation would not be any more effective at communicating the underlying ideas; in fact, if anything, it will serve to distract from them and make a greater barrier to entry when someone is trying to learn the material for the first time.

I'd argue that notation should be judged by three standards: (1) its clarity, (2) its rigor, and (3) whether it somehow invites the reader to make a mistake. On metric (1), I strongly prefer the current "abuses" of notation, particularly when considering this from the perspective of someone who doesn't already understand it. On metric (2), I guess you could give a tiny edge to the alternatives you propose... but not really. If, for instance, you were trying to encode these concepts into a programming language, it would be trivial to fill in the details from the existing notation. And (3) is a wash; there is no meaningful opportunity for error created by the notation in either case.

In short: Notation is purely a construct, and to worship at the altar of pedantry at the cost of clarity (and brevity, which is a legitimate facet of clarity) is misguided.

EDIT: I fear that this may have been more a rant than an answer to the question. So, to be sure that it's both: no, these examples should not be regarded as an abuse of notation. Rather, they're useful and harmless conventions, like all good notation is.

EDIT #2: After reflecting on GitGud's comment, I think they are right; it's hard for me to say with a straight face that (particularly the second example in the original post) is not an abuse of notation. Really, Patrick Stevens already gave the best answer to the original question, and I'm glad it has been upvoted as many times as it has. However, these abuses of notation should, if anything, be regarded as (1) very mild abuses, and (2) useful, perhaps even important, conventions.

A function $f:\>X\to Y$ takes elements $x\in X$ as input and produces for each such $x$ an output value $y\in Y$. In so far any function is "unary".

Now it so happens that in many cases the domain $X$ is a cartesian product $X=R^n$, such that we need $n$ pieces of $R$-data in order to specify a single point $x\in X$, i.e., we write $x=(x_1,x_2,\ldots, x_n)$. The function value $f(x)$ should thus be noted as $f((x_1,x_2,\ldots, x_n))$, but in daily practice one strips one set of parentheses off. Note that in the Mathematica language one writes $f\bigl[\{x_1,\ldots, x_n\}\bigr]$.

Note that in many cases such a "multivariable" function depends on $x$ not in a "cloudy" way. Instead this dependence is clearly structured along the factorization of $X$, as in $f(x,y,z):=(x^2+y^2)e^z$. The "variables" $x$, $y$, $z$ then obtain their own personality and are not just tools to address the intended domain point ${\bf p}$.

Unfortunately nobody could tell me in all those years what a "variable" is in analysis.

If we are going to play the game as you want, your writing has serious issues. For starters, $\mathbb R$ is a set, but then you are using this symbol $+$ which you haven't defined. You also have the symbol $83xy$, which is not clear what it means. It looks like you are abusing notation and writing multiplication of real numbers (that you never said you were using) by juxtaposition; so, is the expression $83xy$ equal to $8\times3\times x\times y$? Maybe not, but in that case you are giving two different meanings to juxtaposition of real numbers, without saying so.

Also, you have those symbols $\pi_1,\pi_2$ that you haven't defined. It looks by your notation that they are functions, but in that case you should express the domain, the codomain, and the rule.

Let reverse the question.

Given a function $f(x,y)=x+y$, what is its domain?

Most people would say $\mathbb R^2$. What would you say?